Discrete Mathematics: An Integrated Approach

This textbook is a culmination of years of work by Dr. Jongsma, recently revised and consolidated for the purpose of use in the MATH 212 course.

2016 Calvin Jongsma

Abstract

Discrete mathematics, you may be disappointed to discover, is not the sort of stuﬀ you talk about discreetly, in secretive whispers behind closed doors. It can be discussed in public by anyone; it has lots of important everyday connections. The term discrete means separate or disconnected; its opposite is continuous. Discrete mathematics deals with non-continuous quantitative phenomena — activities as mundane as counting objects and as arcane as ana-lyzing the output of a Turing machine. It also includes familiar topics like functions, but it explores their structural algebraic features and examines them with respect to computational complexity; matters connected to continuity or diﬀerentiability are left for analytic geometry and calculus.

As a ﬁeld of mathematics, discrete mathematics is both very old and very new. Counting and basic arithmetic, of course, go back to pre-historic times. The rise of discrete mathematics in recent decades, however, is a result of the increased power and ongoing development of computer technology. Concepts and results in discrete mathematics provide the mathematical foundation for computer science. In return, computer science supplies crucial stimuli and resources for important contemporary developments and applications in discrete mathematics.

Discrete mathematics is a ﬁeld every computer scientist should know about, and one that today’s mathematicians should be familiar with as well. This textbook is written for students in both majors. While the primary focus here is on mathematics, we will investigate topics that are crucial for computer science as well.